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A *-compactification of a T 1 quasi-uniform space (X,U ) is a compact T 1 quasi-uniform space (Y,V ) that has a T( V *)-dense subspace quasi-isomorphic to (X,U ), where V * denotes the coarsest uniformity finer than V .In this paper we characterize all Wallman type compactifications of a T 1 topological space in terms of the *-compactification of its point symmetric totally bounded transitive compatible quasi-uniformities. We deduce that the *-compactification of the Pervin quasi-uniformity of any normal T 1 topological space X is exactly the Stone-Cech compactification of X. We also obtain a characterization of those Hausdorff compactifications of a given space, which are of Wallman type.
Summary
Given a quasi-uniform space (X,U), we study its Hausdorff quasi-uniformity UH on the set P0(X) of nonempty subsets of the set X. In particular we are concerned with the question whether at a certain finite stage iterations of the described Hausdorff hyperspace construction applied to two distinct quasi-uniformities on X will necessarily lead to hyperspaces carrying distinct induced topologies.
Abstract
On every approach space X, we construct a compatible quasi-uniform gauge structure which turns out to be at the same time the coarsest functorial structure and the finest compatible totally bounded one. Based on the analogy with the classical Császár-Pervin quasi-uniform space, we call this the “Császár-Pervin” quasi-uniform gauge space. By means of the bicompletion of this Császár-Pervin quasi-uniform gauge space of a T 0 approach space X, we succeed in constructing the sobrification of X.
Abstract
V. Gregori and S. Romaguera [17] obtained an example of a fuzzy metric space (in the sense of A. George and P. Veeramani) that is not completable, i.e. it is not isometric to a dense subspace of any complete fuzzy metric space; therefore, and contrary to the classical case, there exist quiet fuzzy quasi-metric spaces that are not bicompletable neither D-completable, via (quasi-)isometries. In this paper we show that, nevertheless, it is possible to obtain solutions to the problem of completion of fuzzy quasi-metric spaces by using quasi-uniform isomorphisms instead of (quasi-)isometries. Such solutions are deduced from a general method, given here, to obtain extension properties of fuzzy quasi-metric spaces from the corresponding ones of the classical theory of quasi-uniform and quasi-metric spaces.
quasi-uniform spaces Acta Math. Hungar. 64 41 – 54 10.1007/BF01873968 . [16] Deák , J. 1995 A
.01.025 . [4] Fletcher , P. Lindgren , W. F. 1982 Quasi-uniform Spaces Dekker New York . [5
